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Gunnar Carlsson

Publications and source records attributed to Gunnar Carlsson.

At least 19 recordsLinked to original sources
arXiv2026

Algebraic K-theory of Geometric Groups

In this paper we introduce a homotopy theoretic technique for proving that the $K$-theoretic assembly map is an equivalence. It is an extension of the methods used to prove split injectivity of the assembly and applies to any geometrically finite group. Our result is that there are two requirements which need to hold. The first is that the assembly map for the group regarded as a metric space is an equivalence. This is a non-equivariant condition and depends only on the coarse type of the word metric on the group. The second is that the group ring satisfies an algebraic coherence condition, which currently can be verified for all known groups for which the split injectivity statement for the assembly holds. The two conditions extend very broadly. In particular, both conditions hold for groups of finite asymptotic dimension. To state the main theorem precisely, given a regular Noetherian ring $A$ of finite global dimension and a group $Γ$ with finite $K(Γ,1)$ and finite asymptotic dimension, we prove that the $K$-theoretic assembly map is an equivalence. Therefore, in all dimensions the $K$-theory of $A[Γ]$ is the group homology of $Γ$ with coefficients in the $K$-theory spectrum of $A$. One of the many geometric consequences of this theorem is vanishing of the Whitehead group of $Γ$.

math.AT↗
arXiv2025

Metric Representations of Network Data

Networks are structures that encode relationships between pairs of elements or nodes. However, there is no imposed connection between these relationships, i.e., the relationship between two nodes can be independent of every other one in the network, and need not be defined for every possible pair of nodes. This is not true for metric spaces, where the triangle inequality imposes conditions that must be satisfied by triads of distances in the space, and these distances must be defined for every pair of nodes. In this paper, we study how to project networks into q-metric spaces, a generalization of metric spaces that encompasses a larger class of structured representations. In order to do this, we encode as axioms two intuitively desirable properties of the mentioned projections. We show that there is only one way of projecting networks onto q-metric spaces satisfying these axioms. Moreover, for the special case of (regular) metric spaces, this method boils down to computing the shortest path between every node and, for the case of ultrametric spaces, it coincides with single linkage hierarchical clustering. Furthermore, we show that the projection method satisfies two properties of practical relevance: optimality, which enables its utilization for the efficient estimation of combinatorial optimization problems, and nestedness, which entails consistency of the structure induced when projecting onto different q-metric spaces. Finally, we illustrate how metric projections can be used to efficiently search networks aided by metric trees.

cs.SI↗
arXiv2025

Topological Data Analysis and Topological Deep Learning Beyond Persistent Homology -- A Review

Topological data analysis (TDA) is a rapidly evolving field in applied mathematics and data science that leverages tools from topology to uncover robust, shape-driven insights in complex datasets. The main workhorse is persistent homology, a technique rooted in algebraic topology. Paired with topological deep learning (TDL) or topological machine learning, persistent homology has achieved tremendous success in a wide variety of applications in science, engineering, medicine, and industry. However, persistent homology has many limitations due to its high-level abstraction, insensitivity to non-topological changes, and reliance on point cloud data. This paper presents a comprehensive review of TDA and TDL beyond persistent homology. It analyzes how persistent topological Laplacians and Dirac operators provide spectral representations to capture both topological invariants and homotopic evolution. Other formulations are presented in terms of sheaf theory, Mayer topology, and interaction topology. For data on differentiable manifolds, techniques rooted in differential topology, such as persistent de Rham cohomology, persistent Hodge Laplacian, and Hodge decomposition, are reviewed. For one-dimensional (1D) curves embedded in 3-space, approaches from geometric topology are discussed, including multiscale Gauss-link integrals, persistent Jones polynomials, and persistent Khovanov homology. This paper further discusses the appropriate selection of topological tools for different input data, such as point clouds, sequential data, data on manifolds, curves embedded in 3-space, and data with additional non-geometric information. A review is also given of various topological representations, software packages, and machine learning vectorizations. Finally, this review ends with concluding remarks.

math.HO↗
arXiv2024

Position: Topological Deep Learning is the New Frontier for Relational Learning

Topological deep learning (TDL) is a rapidly evolving field that uses topological features to understand and design deep learning models. This paper posits that TDL is the new frontier for relational learning. TDL may complement graph representation learning and geometric deep learning by incorporating topological concepts, and can thus provide a natural choice for various machine learning settings. To this end, this paper discusses open problems in TDL, ranging from practical benefits to theoretical foundations. For each problem, it outlines potential solutions and future research opportunities. At the same time, this paper serves as an invitation to the scientific community to actively participate in TDL research to unlock the potential of this emerging field.

cs.LG↗
arXiv2024

Equivariant Algebraic K-Theory and Derived completions II: the case of Equivariant Homotopy K-Theory and Equivariant K-Theory

In the mid 1980s, while working on establishing completion theorems for equivariant Algebraic K-Theory similar to the well-known completion theorems for equivariant topological K-theory, the late Robert Thomason found the strong finiteness conditions that are required in such theorems to be too restrictive. Then he made a conjecture on the existence of a completion theorem for equivariant Algebraic G-theory, for actions of linear algebraic groups on schemes that holds without any of the strong finiteness conditions that are required in such theorems proven by him. In an earlier work by the first two authors, we solved this conjecture by providing a derived completion theorem for equivariant G-theory. In the present paper, we provide a similar derived completion theorem for the homotopy Algebraic K-theory of equivariant perfect complexes, on schemes that need not be regular. Our solution is broad enough to allow actions by all linear algebraic groups, irrespective of whether they are connected or not, and acting on any normal quasi-projective scheme of finite type over a field, irrespective of whether they are regular or projective. This allows us therefore to consider the Equivariant Homotopy Algebraic K-Theory of large classes of varieties like all toric varieties (for the action of a torus) and all spherical varieties (for the action of a reductive group). With finite coefficients invertible in the base fields, we are also able to obtain such derived completion theorems for equivariant algebraic K-theory but with respect to actions of diagonalizable group schemes. These enable us to obtain a wide range of applications, several of which are also explored.

math.AG↗
arXiv2024

Equivariant Algebraic K-Theory and Derived completions III: Applications

In the present paper, we discuss applications of the derived completion theorems proven in our previous two papers. One of the main applications is to Riemann-Roch problems for forms of higher equivariant K-theory, which we are able to establish in great generality both for equivariant G-theory and equivariant homotopy K-theory with respect to actions of linear algebraic groups on normal quasi-projective schemes over a given field. We show such Riemann-Roch theorems apply to all toric and spherical varieties. We also obtain Lefschetz-Riemann-Roch theorems involving the fixed point schemes with respect to actions of diagonalizable group schemes. We also show the existence of certain spectral sequences that compute the homotopy groups of the derived completions of equivariant G-theory starting with equivariant Borel-Moore motivic cohomology.

math.AG↗
arXiv2024

Motivic and Étale Spanier-Whitehead duality and the Becker-Gottlieb transfer

In this paper, we develop a theory of Becker-Gottlieb transfer based on Spanier-Whitehead duality that holds in both the motivic and étale settings for smooth quasi-projective varieties in as broad a context as possible: for example, for varieties over non-separably closed fields in all characteristics, and also for both the étale and motivic settings. In view of the fact that the most promising applications of the traditional Becker-Gottlieb transfer has been to torsors and Borel-style equivariant cohomology theories, we focus our applications to motivic cohomology theories for torsors as well as Borel-style equivariant motivic cohomology theories, both defined with respect to motivic spectra. We obtain several results in this direction, including a stable splitting in generalized motivic cohomology theories. Various further applications will be discussed in forthcoming papers.

math.AG↗
arXiv2023

Current Topological and Machine Learning Applications for Bias Detection in Text

Institutional bias can impact patient outcomes, educational attainment, and legal system navigation. Written records often reflect bias, and once bias is identified; it is possible to refer individuals for training to reduce bias. Many machine learning tools exist to explore text data and create predictive models that can search written records to identify real-time bias. However, few previous studies investigate large language model embeddings and geometric models of biased text data to understand geometry's impact on bias modeling accuracy. To overcome this issue, this study utilizes the RedditBias database to analyze textual biases. Four transformer models, including BERT and RoBERTa variants, were explored. Post-embedding, t-SNE allowed two-dimensional visualization of data. KNN classifiers differentiated bias types, with lower k-values proving more effective. Findings suggest BERT, particularly mini BERT, excels in bias classification, while multilingual models lag. The recommendation emphasizes refining monolingual models and exploring domain-specific biases.

cs.CY↗
arXiv2022

Evasion paths by homotopy limits

The evasion paths problem asks when a dynamically changing space can be navigated: imagine guards are patrolling a region, for instance, and we need to stay outside their view. We use the Bousfield-Kan spectral sequence for homotopy inverse limits as a proxy for calculating the homotopy groups of the space of evasion paths. This gives a complete calculation of the number of paths (up to homotopy) for spaces we call ``0-sectional." We give a characterization such spaces, and conclude the paper by reframing the calculations in terms of the covered region.

math.AT↗
arXiv2022

Approximating Absolute Galois Groups

This paper formulates a group condition which is enjoyed by absolute Galois groups, and which guarantees that profinite groups satisfying the condition can be approximated as an inverse limit of groups which are profinite analogues of Bieberbach groups. The condition is a key ingredient in the study of the "representational assembly" approach to descent problems in algebraic K-theory.

math.KT↗
arXiv2021

Robust Hierarchical Clustering for Directed Networks: An Axiomatic Approach

We provide a complete taxonomic characterization of robust hierarchical clustering methods for directed networks following an axiomatic approach. We begin by introducing three practical properties associated with the notion of robustness in hierarchical clustering: linear scale preservation, stability, and excisiveness. Linear scale preservation enforces imperviousness to change in units of measure whereas stability ensures that a bounded perturbation in the input network entails a bounded perturbation in the clustering output. Excisiveness refers to the local consistency of the clustering outcome. Algorithmically, excisiveness implies that we can reduce computational complexity by only clustering a subset of our data while theoretically guaranteeing that the same hierarchical outcome would be observed when clustering the whole dataset. In parallel to these three properties, we introduce the concept of representability, a generative model for describing clustering methods through the specification of their action on a collection of networks. Our main result is to leverage this generative model to give a precise characterization of all robust -- i.e., excisive, linear scale preserving, and stable -- hierarchical clustering methods for directed networks. We also address the implementation of our methods and describe an application to real data.

cs.LG↗
arXiv2021

Evaluating the Disentanglement of Deep Generative Models through Manifold Topology

Learning disentangled representations is regarded as a fundamental task for improving the generalization, robustness, and interpretability of generative models. However, measuring disentanglement has been challenging and inconsistent, often dependent on an ad-hoc external model or specific to a certain dataset. To address this, we present a method for quantifying disentanglement that only uses the generative model, by measuring the topological similarity of conditional submanifolds in the learned representation. This method showcases both unsupervised and supervised variants. To illustrate the effectiveness and applicability of our method, we empirically evaluate several state-of-the-art models across multiple datasets. We find that our method ranks models similarly to existing methods. We make ourcode publicly available at https://github.com/stanfordmlgroup/disentanglement.

stat.ML↗
arXiv2021

Topological Deep Learning

This work introduces the Topological CNN (TCNN), which encompasses several topologically defined convolutional methods. Manifolds with important relationships to the natural image space are used to parameterize image filters which are used as convolutional weights in a TCNN. These manifolds also parameterize slices in layers of a TCNN across which the weights are localized. We show evidence that TCNNs learn faster, on less data, with fewer learned parameters, and with greater generalizability and interpretability than conventional CNNs. We introduce and explore TCNN layers for both image and video data. We propose extensions to 3D images and 3D video.

cs.LG↗
arXiv2021

Persistent and Zigzag Homology: A Matrix Factorization Viewpoint

Over the past two decades, topological data analysis has emerged as a field of applied mathematics with new applications and algorithmic developments appearing rapidly. Two fundamental computations in this field are persistent homology and zigzag homology. In this paper, we show how these computations in the most general case reduce to finding a canonical form of a matrix associated with a type A quiver representation, which in turn can be computed using factorizations of associated matrices. We show how to use arbitrary induced maps on homology for computation, providing a framework that goes beyond the capabilities of existing software for topological data analysis. Furthermore, this framework offers multiple opportunities for parallelization which have not been previously exploited. We provide several examples of the utility of this framework, demonstrate parallel speedups, and report on significant improvements in comparison to existing software.

math.AT↗
arXiv2020

The space of sections of a smooth function

Given a compact manifold $X$ with boundary and a submersion $f : X \rightarrow Y$ whose restriction to the boundary of $X$ has isolated critical points with distinct critical values and where $Y$ is $[0,1]$ or $S^1$, the connected components of the space of sections of $f$ are computed from $π_0$ and $π_1$ of the fibers of $f$. This computation is then leveraged to provide new results on a smoothed version of the evasion path problem for mobile sensor networks: From the time-varying homology of the covered region and the time-varying cup-product on cohomology of the boundary, a necessary and sufficient condition for existence of an evasion path and a lower bound on the number of homotopy classes of evasion paths are computed. No connectivity assumptions are required.

math.AT↗
arXiv2020

A Topology Layer for Machine Learning

Topology applied to real world data using persistent homology has started to find applications within machine learning, including deep learning. We present a differentiable topology layer that computes persistent homology based on level set filtrations and edge-based filtrations. We present three novel applications: the topological layer can (i) regularize data reconstruction or the weights of machine learning models, (ii) construct a loss on the output of a deep generative network to incorporate topological priors, and (iii) perform topological adversarial attacks on deep networks trained with persistence features. The code (www.github.com/bruel-gabrielsson/TopologyLayer) is publicly available and we hope its availability will facilitate the use of persistent homology in deep learning and other gradient based applications.

cs.LG↗
arXiv2020

Persistent Homology and Applied Homotopy Theory

This paper is a survey of persistent homology, primarily as it is used in topological data analysis. It includes the theory of persistence modules, as well as stability theorems for persistence barcodes, generalized persistence, vectorization of persistence barcodes, as well as some applications.

math.AT↗